1. Math project 12-54 ES-A

2. Done by: • AzzaEissa • Alanoudhamed • Salama mohammed

3. Differentiation Project • In this project you will work with one of the most important processes of calculus . You will apply the rules you have learned to find answers of real life problems that have faced the aircraft engineers during a full Airbus journey.

4. History • History Before starting the A380 project both Airbus and Boeing had focused on cornering the very-large-airliner market. Airbus and Boeing had worked together on a study investigating a 600+ seat aircraft called the Very Large Commercial Transport, but this cooperation did not last long. Although both manufacturers issued various statements, the unspoken consensus was that there was probably room for only one maker to be profitable in the 600 to 800 seat market segment. Both knew the risk of splitting a niche market. • The simultaneous debut of the Douglas DC-10 and the Lockheed Tristar had demonstrated this: either aircraft could technically fill the gap between the Douglas DC-8 and the Boeing 747, but the market could only sustain one of the two. Eventually both companies left the civil airliner market. Boeing and Airbus decided to enter the new 600 seat market. • Boeing initially had the upper-hand. The 747, though designed in the 1960s, was popular and larger than Airbus' largest jet, the A340. For many airlines the extra size of the 747 made it a "must buy" for their highest density routes and the lower costs of a common fleet led carriers to buy additional Boeing aircraft. Boeing was considering a New Large Aircraft to replace the 747 and acquired McDonnell Douglas and their cancelled MD-12 design. Boeing studied the concept of the 747X, a version of the 747 with the forebody "hump" extended towards the rear for more passenger room, before dropping the concept. Development of the "A3XX" began in June, 1994. In 2001, it was re-branded the A380 with the announcement of Singapore Airlines as the launch customer

6. Airbus A380 characteristics. • The A380-800 is a subsonic, very long range, very high capacity, civil transport aircraft. The A380-800 offers several payload capabilities ranging from 400 passengers in a very comfortable multiclass configuration, up to 853 passengers in an all economy class configuration. Designed in close collaboration with major airlines, airports and airworthiness authorities, the A380 is the most advanced, spacious and productive aircraft in service setting a new standard in air travel and environmental efficiency.

7. How long it takes an A 380 to takeoff. Takeoff velocity = 78 m/s Takeoff acceleration = 1 m/s2 Therefore, it takes from the airbus A380 78 seconds to takeoff. a = Δv/Δt 1 = 78/Δt Δt = 78s

8. What is the takeoff velocity? • It varies a bit depending on things such as winds, take-off weight and air pressure (runway altitude), but under typical conditions take-off speed is around 150 knots (280km/h or 170mph). • Takeoff velocity = 78 m/s

9. What is the takeoff distance,i.e. the runway length that will be crossed before it takes off? Δv = Δd/Δt 78 = Δd/78 Δd = 6084 m Therefore, it takes from the airbus 6084 meter to takeoff.

10. Waiting at the airport • The average number of airplanes waiting in a modern airport to take off can be modeled by the function where is a x2/2(1-x) quantity between 0 and 1 known as the traffic intensity. Find the rate of change of the number of airplanes waiting with respect to the traffic intensity when 𝑥=0.1and 𝑥=0.7 W(x)=x2/(2-2x) u’ = 2x v’ = -2 W’(x) = (u'v-v'u)/v2 = 2x(2-2x)-(-2)(x2)/(2-2x)2 = 4x-4x2+2x2/(2-2x)2 = -2x2+4x/(2-2x)2 When x = 0.1  w’(0.1) = -2(0.1)2+4(0.1)/(2-2(0.1))2 = 19/162 When x = 0.7 w’(0.7) = -2(0.7)2+4(0.7)/(2-2(0.7))2 = 9118

11. Taking-off Distance and Time Suppose that the distance an Airbus travels along a runway before taking off is given by meter d(t) = (10/9)t2measured from the starting point, and 𝑡𝑡 is measured in seconds from the time the brakes are released. The aircraft will become airborne when its speed reaches 72 meters per second. How long will it take to become airborne, and what distance will it travel in that time? D(t) = (10/9)t2 D’(t) = 2(10/9)t D’(t) = (20/90)t (90/20) x 7.2 =t t- 32.4s D(t) = (10/9)t2 D(32.4) = (10/9) (32.4)2 D(32.4) = 1166.4 m

12. Speed of an airplane by Radar • If this airplane was flying on a flight path that would take it directly over a radar station, as shown in figure below. If “s” was decreasing at a rate of 400 miles per hour when “s=10miles” what is the speed of the plane?

13. Aircraft Glide Path Now the Airbus starts descent from an altitude of 1 mile, 4 miles west of the runway. Find the cubic function f(x) = ax3 + bx2 + cx + d on the interval −that describes a smooth glide path for the landing. The function you founded models the glide path of the plane. When would the plane be descending at the greatest rate?

14. Aircraft Glide Path F(-4)= -64a+16b-4c+d 1=-64a+16(6a)-4(0)+0 1=-64a+96a 1=32a 132=3232a a=132 = 0.03125. a=132 b6=132 b=316=0.1875. d=0 , c=0 , a=132 = 0.03125 , b=316=0.1875. F(x)= 132x3+316x2 At (0,0).

15. Read about Doppler effect of moving objects.You may Use these link to know more about Doppler effect • • If the frequency of an airbus heard by a stationary witness on runway; frequency noticed by the witness is the frequency of the source, the velocity of the waves, the velocity of the listener, and the velocity of the source. • • Find an equation of the in term of. Then find the rate of change of with respect to , In two cases once when airbus is approaching at velocity of 200km/hr, and the other when its moving away at a velocity 200km/hr

16. The End